When "too" isn't agreeing... what is it doing?

Hi, odd question here. Doing some research and discourse analysis (coding different types of speech acts), and came across this tricky bit of semantics that I could use help and/or varied opinions about.

So when someone gives an opinion such as "I love the summer!" you might say "me, too!" or "D&D is so much fun!" "Yeah, I like it, too!" and this is all fine and dandy because you're agreeing with an opinion. We could replace both of those responses with "I agree!" and it would make perfect sense semantically. However something changes when an opinion is not at the center of the discourse. Take a simple fact for example:

"I'm a doctor." "Me, too."

In this case, switching the response to "I agree." Does not have the same meaning. Responding with "I agree" in such a case would imply that you also think that person A is a doctor, and it speaks nothing of what you yourself are. So if the function of "Me, too" is not "agreeing" in this case, what IS it's function? What would you call this? Conforming? Concurring? Cohering? What word (verb preferably) expresses the action of "expressing that your condition/status is the same as another's". Does such a word even exist?